Dear M:

You wrote:

In this sentence, what do the ‘thats’ mean and stand for?

“I doubt that that was the idea.”

I know the second ‘that’ refers to the object which was doubted, but what about the first ‘that’, and is this sentence correct.

The first “that” is a relative pronouns. The first one is relating the main clause to the subordinate clause. We use “that” like that many times. In modern English, it is optional, but it is usually written in any formal usage.

Examples:
He said that he would be late.
He said he would be late. (use is optional)

The second “that” is a demonstrative pronoun which is acting as the subject of the subordinate clause. “That” in this sense shows something you are pointing to or pointing out, but it is more distant than “this.” If you can replace the “that” with a “this” and the sentence still makes sense, then you know you have a demonstrative adjective: “I doubt that this was the idea.”

Yes, the sentence is correct. In many cases we would drop the first “that,” to say, “I doubt that was the idea,” but it means the same thing.

Basically, “that” has two different uses and meanings–one as a demonstrative pronoun or adjective (“this” vs. “that”) and one as a relative pronoun (connecting clauses). In this case you have one of each.